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## Homework Statement

Let [itex]f_{n}(x)=\frac{-x^2+2x-2x/n+n-1+2/n-1/n^2}{(n ln(n))^2}[/itex]

Prove [itex]f(x) = \sum^{\infty}_{n=1} f_{n}(x)[/itex] is well defined and continuous on the interval [0,1].

## Homework Equations

In a complete normed space, if [itex]\sum x_{k}[/itex]converges absolutely, then it converges.

## The Attempt at a Solution

Working in a complete normed space [itex](C[0,1], || . ||_{∞})[/itex],

consider the real series [itex]\sum^{∞}_{n=1}||f_{n}||_{∞}=\sum^{∞}_{n=1} sup <f_{n}(x) : x\in[0,1]> [/itex]

It just remains to show that [itex]\sum^{∞}_{n=1}|f_{n}|[/itex] converges, but I can't seem to figure out how. Could anyone help me out here?

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